PROC GLM for Quadratic Least Squares Regression

In polynomial regression, the values of a dependent variable
(also called a response variable) are described or predicted
in terms of polynomial terms involving one or more independent or explanatory variables.
An example of quadratic regression in PROC GLM follows. These data are
taken from Draper and Smith (1966, p. 57). Thirteen
specimens of 90/10 Cu-Ni alloys are tested in a
corrosion-wheel setup in order to examine corrosion.
Each specimen has a certain iron content. The
wheel is rotated in salt sea water at 30 ft/sec for
60 days. Weight loss is used to quantify the corrosion.
The fe variable represents the iron content, and the loss variable denotes the
weight loss in milligrams/square
decimeter/day in the following DATA step.

The GPLOT procedure is used to request a scatter plot of
the response variable versus the independent variable.

symbol1 c=blue;
proc gplot;
plot loss*fe / vm=1;
run;

The plot in Figure 30.3 displays a strong negative
relationship between iron content and corrosion resistance, but it is
not clear whether there is curvature in this relationship.

Figure 30.3: Plot of LOSS vs. FE

The following statements fit a quadratic regression model
to the data. This enables you to estimate the linear relationship
between iron content
and corrosion resistance and
test for the presence of
a quadratic component. The intercept is automatically
fit unless the NOINT option is specified.

proc glm;
model loss=fe fe*fe;
run;

The CLASS statement is omitted because
a regression line is being fitted. Unlike PROC REG,
PROC GLM allows polynomial terms in the MODEL statement.

Regression in PROC GLM

The GLM Procedure

Number of observations

13

Figure 30.4: Class Level Information

The preliminary information in Figure 30.4
informs you that the GLM procedure has been invoked
and states the number of observations in the data set.
If the model involves classification variables, they
are also listed here, along with their levels.

Figure 30.5 shows the overall ANOVA table and some simple
statistics. The degrees of freedom can be used to check that
the model is correct and that the data have been
read correctly. The Model degrees of freedom for a
regression is the number of parameters in the model minus
1. You are fitting a model with three parameters in this case,

so the degrees of freedom are 3-1=2.
The Corrected Total degrees of freedom are always
one less than the number of observations used in the
analysis.

Regression in PROC GLM

The GLM Procedure

Dependent Variable: loss

Source

DF

Sum of Squares

Mean Square

F Value

Pr > F

Model

2

3296.530589

1648.265295

164.68

<.0001

Error

10

100.086334

10.008633

Corrected Total

12

3396.616923

R-Square

Coeff Var

Root MSE

loss Mean

0.970534

2.907348

3.163642

108.8154

Figure 30.5: ANOVA Table

The R2 indicates that the model accounts for
97% of the variation in LOSS. The coefficient of
variation (C.V.), Root MSE (Mean Square for Error),
and mean of the dependent variable are also listed.

The overall F test is significant (F=164.68,
p<0.0001), indicating that the model as a whole
accounts for a significant amount of the variation
in LOSS. Thus, it is appropriate to proceed to testing the
effects.

Figure 30.6 contains tests of effects and
parameter estimates. The latter are displayed by default when
the model contains only continuous variables.

Regression in PROC GLM

The GLM Procedure

Dependent Variable: loss

Source

DF

Type I SS

Mean Square

F Value

Pr > F

fe

1

3293.766690

3293.766690

329.09

<.0001

fe*fe

1

2.763899

2.763899

0.28

0.6107

Source

DF

Type III SS

Mean Square

F Value

Pr > F

fe

1

356.7572421

356.7572421

35.64

0.0001

fe*fe

1

2.7638994

2.7638994

0.28

0.6107

Parameter

Estimate

Standard Error

t Value

Pr > |t|

Intercept

130.3199337

1.77096213

73.59

<.0001

fe

-26.2203900

4.39177557

-5.97

0.0001

fe*fe

1.1552018

2.19828568

0.53

0.6107

Figure 30.6: Tests of Effects and Parameter Estimates

The t tests provided are equivalent to the Type III
F tests. The quadratic term
is not significant (F=0.28, p=0.6107; t=0.53, p=0.6107)
and thus can be removed from the model;
the linear term is significant (F=35.64, p=0.0001;
t=-5.97, p=0.0001). This suggests that there is indeed
a straight line relationship between loss and fe.

Fitting the model without the quadratic term provides
more accurate estimates for and .PROC GLM allows only one MODEL statement per invocation
of the procedure, so the PROC GLM statement must be issued again.
The statements used to fit the linear model are

proc glm;
model loss=fe;
run;

Figure 30.7 displays the output produced by these statements.
The linear term is still significant (F=352.27, p<0.0001). The
estimated model is now